Question

Cho P = \(\frac{x^2+x}{x^2-2x+1}\left(\frac{x+1}{x}-\frac{1}{1-x}+\frac{2-x^2}{x^2-x}\right)\)

a) Rút gọn P

b) Tìm x nguyên để P là số nguyên

c) Với x>1 tìm GTNN của P

Answer 1

a)P=\(\frac{x^2+x}{x^2-2x+1}\left(\frac{x+1}{x}-\frac{1}{1-x}+\frac{2-x^2}{x^2-x}\right)\) ĐKXĐ: x \(\ne\) 1;0

P= \(\frac{x\left(x+1\right)}{\left(x-1\right)^2}\left(\frac{\left(x+1\right)\left(x-1\right)}{x\left(x-1\right)}+\frac{x}{x\left(x-1\right)}+\frac{2-x^2}{x\left(x-1\right)}\right)\)

P=\(\frac{x\left(x+1\right)}{\left(x-1\right)^2}\left(\frac{x^2-1+x+2-x^2}{x\left(x-1\right)}\right)\)

P= \(\frac{x\left(x+1\right)}{\left(x-1\right)^2}.\frac{x+1}{x\left(x-1\right)}\)

P= \(\frac{\left(x+1\right)^2}{\left(x-1\right)^3}\)